\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 118, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2009/118\hfil Antiplane frictional contact]
{Analysis of electro-viscoelastic antiplane contact problem
with total slip rate dependent friction}
\author[M. Dalah\hfil EJDE-2009/118\hfilneg]
{Mohamed Dalah}
\dedicatory{To the memory of Professor Y. B. Boukhadra}
\address{Mohamed Dalah \newline
Laboratoire Mod\'elisation Math\'ematiques et Simulation (LMMS) \\
D\'{e}partement de \newline
Mat\'ematiques, Facult\'{e} des Sciences \\
Universit\'e Mentouri de Constantine \\
Route Ain El-Bey Zerzara, 25 000 Constantine, Algiria}
\email{mdalah17@yahoo.fr}
\thanks{Submitted March 3, 2009. Published September 27, 2009.}
\subjclass[2000]{74M10, 74F15, 74G25, 49J40}
\keywords{Antiplane problem; total slip rate dependent friction law;
\hfill\break\indent electro-viscoelastic law; fixed point;
weak solution; variational inequality; Tresca's friction law}
\begin{abstract}
We consider a mathematical model which describes the antiplane
shear deformation of a cylinder in frictional contact with a rigid
foundation. The contact is bilateral and is modelled with a total
slip rate dependent friction law. The material is assumed to be
electro-viscoelastic and the foundation is assumed to be
electrically conductive. First, we describe the classical
formulation for the antiplane problem and we give the
corresponding variational formulation which is given by a system
coupling an evolutionary variational equality for the displacement
field and a time-dependent variational equation for the electric
potential field. Then we prove the existence of a unique weak
solution to the model. The proof is based on arguments of
variational inequalities and by using the Banach fixed-point
Theorem.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks
\section{Introduction}\label{s:Int}
The contact between deformable bodies is a phenomenon frequently
found in industry and in everyday life. The contact of the
breaking pads with the wheel, of the tire with the road and the
piston with the skirt are just simples examples. Considerable
progress has been achieved recently in modelling, mathematical
analysis and numerical simulations of various contact processes
and, as a result, a general Mathematical Theory of Contact
Mechanics (MTCM) is currently maturing. It is concerned with the
mathematical structures which underlie general contact problems
with different constitutive laws (i.e., different materials),
varied geometries and settings, and different contact conditions,
see for instance \cite {HS,SST,SE1} and the references therein.
The reason is that, owing to the inherent complicated nature,
contact phenomena are modelled by difficult nonlinear problems,
which explains the slow progress in their mathematical analysis.
Piezoelectric materials are characterized by the coupling between
the mechanical and electrical properties. This coupling leads to
the appearance of electric potential when mechanical stress is
present and, conversely, mechanical stress is generated when
electric potential is presented. The first effect is used in
mechanical sensors and the reverse effect is used in actuators, in
engineering control equipment. Piezoelectric materials for which
the mechanical properties are elastic are called electro-elastic
materials and those for which the mechanical properties are
viscoelastic are called electro-viscoelastic material's. General
models for electro-viscoelastic materials can be found in \cite
{AMS} In all these references the foundation was assumed to be
electrically insulated. Antiplane problems \cite
{AKS,Ch.Han.Sof,ChO,RSS1,HS} for piezoelectric materials were
considered in \cite {TH.AN,AwSS,J.Eck,Ro.Shi.So,Tallec,HM,SE2}. We
rarely actually load piezoelectric bodies so as to cause them to
deform in antiplane shear; however, the governing equations and
boundary conditions for antiplane shear problems involving
piezoelectric materials are beautifully simple and the solution
has many of the features of the more general case and may help us
to solve the more complex problem too.
In this paper, as in \cite {Ro.Shi.So,Tallec,HM,SE2,SDA}; there a
model for the antiplane contact of an electro-elastic cylinder was
considered under the assumption that the foundation is
electrically conductive; the variational formulation of the model
was derived and the existence of a unique solution to the model
was proved by using arguments of evolutionary variational
inequalities. Unlike \cite {AwSS,Ma.Mo.Sof1,Ro.Shi.So,HS}, in the
present paper we consider a quasistatic contact problem between a
rigid foundation and a cylinder. This problem is considered to be
antiplane, i.e. the displacements parallel to the generators of
the cylinder and is independent to the axial coordinate. Our
interest is to describe a simple physical process in which both
frictional contact, viscosity and piezoelectric effects are
involved, and to show that the resulting model leads to a
well-posed mathematical problem. Taking into account the
frictional contact between a viscous piezoelectric body and an
electrically conductive foundation in the study of an antiplane
problem leads to a new and interesting mathematical model which
has the virtue of relative mathematical simplicity without loss of
essential physical relevance.
Our paper is structured as follows. In section \ref{s:2} we
present the model of the antiplane frictional contact of an
electro-viscoelastic cylinder. In section \ref{s:3} we introduce
the notation, list the assumption on problem's data, derive the
variational formulation of the problem and state our main
existence and uniqueness result; i.e., Theorem \ref{t:1}. The
proof of this result is carried out in several steps in Section
\ref{s:4} and is based on the argument of evolutionary variational
inequalities and Banach's fixed point.
\section{The antiplane contact problem} \label{s:2}
We consider a piezoelectric body $\mathcal{B}$ identified with a
region in $\mathbb{R}^3$ it occupies in a fixed and undistorted
reference configuration. We assume that $\mathcal{B}$ is a
cylinder with generators parallel to the $x_3$-axes with a
cross-section which is a regular region $\Omega$ in the $x_1$,
$x_2$-plane, $Ox_1x_2x_3$ being a Cartesian coordinate system. The
cylinder is assumed to be sufficiently long so that the end
effects in the axial direction are negligible. Thus,
$\mathcal{B}=\Omega\times (-\infty,+\infty)$. The cylinder is
acted upon by body forces of density $\mathbf{f}_0$ and has volume
free electric charges of density $q_0$. It is also constrained
mechanically and electrically on the boundary. To describe the
boundary conditions, we denote by $\partial \Omega=\Gamma$ the
boundary of $\Omega$ and we assume a partition of $\Gamma$ into
three open disjoint parts $\Gamma_1$, $\Gamma_2$ and $\Gamma_3$,
on the one hand, and a partition of $\Gamma_1\cup\Gamma_2$ into
two open parts $\Gamma_a$ and $\Gamma_b$, on the other hand. We
assume that the one-dimensional measure of $\Gamma_1$ and
$\Gamma_a$, denoted $\mathop{\rm meas}\Gamma_1$ and $\mathop{\rm
meas}\Gamma_a$, are positive. The cylinder is clamped on
$\Gamma_1\times (-\infty,+\infty)$ and therefore the displacement
field vanishes there. Surface tractions of density $\mathbf{f}_2$
act on $\Gamma_2\times (-\infty,+\infty)$. We also assume that the
electrical potential vanishes on $\Gamma_a\times
(-\infty,+\infty)$ and a surface electrical charge of density
$q_b$ is prescribed on $\Gamma_b\times (-\infty,+\infty) $. The
cylinder is in contact over $\Gamma_3\times (-\infty,+\infty)$
with a conductive obstacle, the so called foundation. The contact
is frictional and is modelled with Tresca's law. We are interested
in the deformation of the cylinder on the time interval $[0,T]$.
Below in this paper the indices $i$ and $j$ denote components of
vectors and tensors and run from $1$ to $3$, summation over two
repeated indices is implied, and the index that follows a comma
represents the partial derivative with respect to the
corresponding spatial variable; also, a dot above represents the
time derivative. We use $\mathcal{S}^3$ for the linear space of
second order symmetric tensors on $\mathbb{R}^3$ or, equivalently, the
space of symmetric matrices of order $3$, and $``\cdot"$,
$\|\cdot\|$ will represent the inner products and the Euclidean
norms on $\mathbb{R}^3$ and $\mathcal{S}^3$; we have:
\begin{gather*}
\mathbf{u} \cdot \mathbf{v} = u_i v_i, \quad
\|\mathbf{v} \| =(\mathbf{v}\cdot\mathbf{v})^{1/2} \quad
\text{for all }\mathbf{u}=(u_i),\; \mathbf{v} =(v_i)\in\mathbb{R}^3,\\
\boldsymbol{\sigma}\cdot\boldsymbol{\tau} = \sigma_{ij} \tau_{ij},
\quad\ \| \boldsymbol{\tau} \| = (\boldsymbol{\tau}\cdot\boldsymbol{\tau}
)^{1/2} \quad \text{for all } \boldsymbol{\sigma}=(\sigma_{ij}),\;
\boldsymbol{\tau}=(\tau_{ij}) \in \mathcal{S}^3.
\end{gather*}
We assume that
\begin{equation}\label{2.1}
\mathbf{f}_0=(0,0,f_0),
\end{equation}
with $f_0=f_0(x_1,x_2,t):\Omega\times[0,T]\to\mathbb{R}$, and
\begin{equation}
\label{2.2} \mathbf{f}_2=(0,0,f_2),
\end{equation}
with $f_2=f_2(x_1,x_2,t):\Gamma_2\times[0,T]\to\mathbb{R}$. The body forces (\ref{2.1})and the surface tractions (\ref{2.2}) would be expected to give rise to a deformation of the elastic cylinder whose displacement, denoted by $\mathbf{u}$, is of the form
\begin{equation}
\label{2.3}\mathbf{u}=(0,0,u),
\end{equation}
with $u=u(x_1,x_2,t):\Omega\times[0,T]\to\mathbb{R}$. Such kind of deformation, associated to a displacement
field of the form (\ref{2.3}), is called an \emph{antiplane
shear}. We assume too that
\begin{gather}\label{2.5}
q_0=q_0(x_1,x_2,t),\\
\label{2.6}q_2=q_2(x_1,x_2,t),
\end{gather}
with $q_0:\Omega\times[0,T]\to\mathbb{R}$ and
$q_2:\Gamma_b\times[0,T]\to\mathbb{R}$.
The electric charges (\ref{2.5}), (\ref{2.6}) would be expected to
give rise to deformations and to electric charges of the piezoelectric
cylinder corresponding to an electric potential field $\varphi$
which is independent on $x_3$ and have
the form
\begin{equation} \label{2.7}
\varphi= \varphi(x_1,x_2,t):\Omega\times[0,T]\to\mathbb{R}.
\end{equation}
The infinitesimal strain tensor, denoted by
$\boldsymbol{\varepsilon}(\mathbf{u})=(\varepsilon_{ij}(\mathbf{u}))$,
is defined by
\begin{gather}\label{2.8}
\varepsilon_{ij}(\mathbf{u})=\frac{1}{2}\,(u_{i,j}+u_{j,i}),\quad
i,j=1,2,3,
\end{gather}
where the index that follows a comma indicates a partial
derivative with respect to the corresponding component of the
spatial variable. Moreover, in the sequel, the convention of
summation upon a repeated index is used.
From (\ref{2.3}) and (\ref{2.8}) it follows that, in the case
of the antiplane problem, the infinitesimal strain tensor becomes
\begin{equation}\label{2.10}
\boldsymbol{\varepsilon}(\mathbf{u})=
\begin{pmatrix}
0&0&\frac{1}{2}u_{,1}\\
0&0&\frac{1}{2}u_{,2}\\
\frac{1}{2}u_{,1}&\frac{1}{2}u_{,2}&0
\end{pmatrix}.
\end{equation}
We also denote by $\mathbf{E}(\varphi)=(E_i(\varphi))$
the electric field and by $\mathbf{D}=(D_i)$ the electric
displacement field
where
\begin{gather}
\label{2.8*}\varepsilon_{ij}(\mathbf{u})=\frac{1}{2}\,(u_{i,j}+u_{j,i}),\\
\label{2.9}
E_i(\varphi)=-\varphi,_i.
\end{gather}
Let $\boldsymbol{\sigma}=(\mathbf{\sigma}_{ij})$ denote the stress field.
We suppose that the material's behavior is modelled by an
electro-viscoelastic constitutive law of the form
\begin{gather}\label{2.11}
\boldsymbol{\sigma}=
2\theta\boldsymbol{\varepsilon}(\dot{\mathbf{u}})+
\zeta\,\mbox{tr}\,\boldsymbol{\varepsilon}(\dot{\mathbf{u}})\,\mathbf{I}
+2\mu\boldsymbol{\varepsilon}
(\mathbf{u})+\lambda\,\mbox{tr}\,\boldsymbol{\varepsilon}(\mathbf{u})\,\mathbf{I}-\mathcal{
E}^*\mathbf{E}(\varphi),\\
\label{2.12}
\mathbf{D}=\mathcal{E}\boldsymbol{\varepsilon}(\mathbf{u})+\alpha\mathbf{E}(\varphi),
\end{gather}
where $\zeta$ and $\theta$ are viscosity coefficients, $\lambda$
and $\mu$ are the Lam\'e coefficients, $\mathop{\rm
tr}\,\boldsymbol{\varepsilon}(\mathbf{u})=\varepsilon_{ii}(\mathbf{u})$,
$\mathbf{I}$ is the unit tensor in $\mathbb{R}^3$, $\alpha$ is the
electric permittivity constant, $\mathcal{E}$ represents the
third-order piezoelectric tensor and $\mathcal{E}^*$ is its
transpose. We assume that
\begin{equation}\label{2.13}
\mathcal{E}\boldsymbol{\varepsilon}=
\begin{pmatrix}
e(\varepsilon_{13}+\varepsilon_{31})\\
e(\varepsilon_{23}+\varepsilon_{32})\\
e\varepsilon_{33}
\end{pmatrix} \quad
\forall\boldsymbol{\varepsilon}=(\varepsilon_{ij})\in\mathcal{S}^3,
\end{equation}
where $e$ is a piezoelectric coefficient. We also assume that the
coefficients $\theta$, $\mu$, $\alpha$ and $e$ depend on the
spatial variables $x_1$, $x_2$, but are independent on the spatial
variable $x_3$. Since
$\mathcal{E}\boldsymbol{\varepsilon}\cdot\mathbf{v}=
\boldsymbol{\varepsilon}\cdot\mathcal{E}^*\mathbf{v}$ for all
$\boldsymbol{\varepsilon}\in \mathcal{S}^3$,
$\mathbf{v}\in\mathbb{R}^3$, it follows from (\ref{2.12}) that
\begin{equation}\label{2.14}
\mathcal{E}^*\mathbf{v}=\begin{pmatrix}
0&0&ev_1\\
0&0&ev_2\\
ev_1&ev_2&ev_3
\end{pmatrix} \quad\forall\mathbf{v}=(v_i)\in\mathbb{R}^3.
\end{equation}
Here and below the dot above represents the derivative with respect
to the time variable. The stress field is given by the matrix
\begin{equation}\label{stfie}
\boldsymbol{\sigma}=\begin{pmatrix}
0&0&\sigma_{13}\\
0&0&\sigma_{23}\\
\sigma_{31}&\sigma_{32}&0
\end{pmatrix}.
\end{equation}
In the antiplane context (\ref{2.3}), (\ref{2.7}), using the
constitutive equations (\ref{2.11}), (\ref{2.12}) and equalities
(\ref{2.13}), (\ref{2.14}) it follows that the stress field and the
electric displacement field are given by
\begin{gather} \label{2.15}
\boldsymbol{\sigma}= \begin{pmatrix}
0&0&\theta\dot{u},_1+\mu u,_1+e\varphi,_1\\
0&0&\theta\dot{u},_2+\mu u,_2+e\varphi,_2\\
\theta\dot{u},_1+\mu u,_1+e\varphi,_1&\theta\dot{u},_2\mu
u,_2+e\varphi,_2&0
\end{pmatrix}, \\
\label{2.16}
\mathbf{D}= \begin{pmatrix}
eu,_1-\alpha\varphi,_1\\
eu,_2-\alpha\varphi,_2\\
0
\end{pmatrix}.
\end{gather}
We assume that the process is mechanically quasistatic and
electrically static and therefore is governed by the equilibrium
equations
\begin{gather} \label{2.17}
\mathop{\rm Div}
\boldsymbol{\sigma}+\mathbf{f}_{0}=\boldsymbol{0},\\
\label{2.18}
D_{i,i}-q_0=0\quad\text{in }\mathcal{B}\times (0,T),
\end{gather}
where $\mathop{\rm Div}\boldsymbol{\sigma}=(\sigma_{ij,j})$
represents the divergence of the tensor field $\boldsymbol{\sigma}$. Thus, keeping in mind (\ref{2.15}), (\ref{2.16}), (\ref{2.3}), (\ref{2.7}), (\ref{2.1}) and (\ref{2.5}), the equilibrium
equations above reduce to the following scalar equations
\begin{gather}
\label{2.19}\mathop{\rm div}(\theta\nabla\dot{u}+\mu\nabla
u)+\mathop{\rm div}(e\nabla\varphi)+f_0=0\quad
\text{in }\Omega\times (0,T),\\
\label{2.20}\mathop{\rm div}(e\nabla
u-\alpha\nabla\varphi)=q_0\quad\mbox{in }\Omega\times (0,T).
\end{gather}
Here and below we use the notation
\begin{gather*}
\mathop{\rm div} \boldsymbol{\tau}=
\tau_{1,1}+\tau_{1,2}\quad\text{for } \boldsymbol{\tau}=
(\tau_1(x_1,x_2,t),\tau_2(x_1,x_2,t)),\\
\nabla v= (v_{,1},v_{,2}),\quad
\partial_{\nu}v=v,_1\,\nu_1+v,_2\,\nu_2\quad\text{for }
v=v(x_1,x_2,t).
\end{gather*}
Recall that, since the cylinder is clamped on
$\Gamma_1\times (-\infty,+\infty)$, the displacement field
vanishes there. Thus (\ref{2.3}) implies
\begin{equation} \label{2.21}
u=0\quad \text{on } \Gamma_1\times (0,T),
\end{equation}
the electrical potential vanishes too on
$\Gamma_a\times (-\infty,+\infty)$; thus (\ref{2.7}) imply that
\begin{gather}\label{2.22}
\varphi=0\quad \text{on } \Gamma_a\times (0,T).
\end{gather}
Let $\boldsymbol{\nu}$ denote the unit normal on
$\Gamma\times(-\infty,+\infty)$. We have
\begin{equation}\label{2.23}
\boldsymbol{\nu}=(\nu_1,\nu_2,0),
\end{equation}
with $\nu_i=\nu_i(x_1,x_2):\Gamma\to\mathbb{R}$, $i=1,2$.
For a vector $\mathbf{v}$ we denote by ${v}_\nu$ and
${\mathbf{v}}_\tau$ its
\emph{normal} and \emph{tangential} components on the boundary, given by
\begin{equation}\label{2.24}
{v}_\nu =\mathbf{v}\cdot\boldsymbol{\nu},\quad {\mathbf{v}}_\tau=
\mathbf{v}-{v}_{\nu}\boldsymbol{\nu},
\end{equation}
respectively. In (\ref{2.24}) and everywhere in this paper
$``\cdot"$ represents the inner product on the space
$\mathbb{R}^3$ ($d = 2, 3$). Moreover, for a given stress field
$\mathbf{\sigma}$ we denote by $\sigma_\nu$ and
$\mathbf{\sigma}_\tau$ the \emph{normal} and the \emph{tangential}
components on the
boundary, respectively; i.e.,
\begin{equation}\label{2.25}
\sigma_\nu=(\boldsymbol{\sigma}\boldsymbol{\nu})\cdot\boldsymbol{\nu},
\quad
\boldsymbol{\sigma}_{\tau}=\boldsymbol{\sigma}\boldsymbol{\nu}
-\sigma_{\nu}\boldsymbol{\nu}.
\end{equation}
From (\ref{2.23}), (\ref{2.15}) and (\ref{2.16}) we deduce that the
Cauchy stress vector and the normal component of the electric
displacement field are given by
\begin{equation} \label{2.26}
\boldsymbol{\sigma}\boldsymbol{\nu}=(0,0,\theta\partial_\nu\dot{u}
+\mu\partial_\nu
u+e\partial_\nu\varphi),\quad
\mathbf{D}\cdot\boldsymbol{\nu}=e\partial_\nu
u-\alpha\partial_\nu\varphi.
\end{equation}
Here and subsequently we use the notations $\partial_\nu
u=u_{,1}\nu_{1}+u_{,2}\nu_{2}$ and $\partial_\nu
\varphi=\varphi_{,1}\nu_{1}+\varphi_{,2}\nu_{2}$.
Keeping in mind the traction boundary condition
$\boldsymbol{\sigma}\boldsymbol{\nu}=\mathbf{f}_2$ on
$\Gamma_2\times(-\infty,\infty)$ and the
electric conditions $\mathbf{D}\cdot\boldsymbol{\nu}=q_2$ on
$\Gamma_b\times(-\infty,\infty)$, it follows from
(\ref{2.2}), (\ref{2.6}) and (\ref{2.26}) that
\begin{gather}\label{2.27}
\theta\partial_\nu\dot{u}+\mu\partial_\nu u+e\partial_\nu\varphi
=f_2 \quad\text{on }\Gamma_2\times(0,T),\\
\label{2.28}e\partial_\nu u-\alpha\partial_\nu\varphi=q_2
\quad\text{on }\Gamma_b\times(0,T).
\end{gather}
We now describe the frictional contact condition on
$\Gamma_3\times(-\infty,+\infty)$. Everywhere in this paper the
notation $|\cdot|$ is used to denote the Euclidean norm on
$\mathbb{R^d}$ ($d=1$ or $3$). First, we remark that
from (\ref{2.3}), (\ref{2.23}) and (\ref{2.24}) we obtain $u_\nu=0$,
which shows that the contact is \emph{bilateral}, i.e. there is no
loss of contact during the process. Using again (\ref{2.3}), (\ref{2.23})
and (\ref{2.24}), we find
\begin{equation}\label{2.29}
\mathbf{u}_\tau=(0,0,u).
\end{equation}
Similarly, from (\ref{2.13}), (\ref{2.23}) and (\ref{2.25})
\begin{equation}\label{2.30}
\boldsymbol{\sigma}_\tau=(0,0,\sigma_\tau).
\end{equation}
where
\begin{equation} \label{2.31}
\sigma_\tau=\theta\partial_\nu\dot{u}+\mu\partial_\nu
u+e\partial_\nu\varphi.
\end{equation}
We assume that the friction is invariant with respect to the $x_3$
axis and for all $t\in [0,T]$ it is modelled by the following
conditions on $\Gamma_3$:
\begin{equation}\label{2.32}
\begin{gathered}
|\boldsymbol{\sigma}_\tau(t)|\leq g(\int_{0}^{t}|\dot{\boldsymbol{u}}_\tau(s)|ds),\\
|\boldsymbol{\sigma}_\tau(t)|0$ and $\mathop{\rm meas}\Gamma_a>0$,
it is well known that $V$ and
$W$ are real Hilbert spaces with the inner products
\[
(u,v)_V=\int_{\Omega}\nabla u\cdot\nabla v\,dx\quad\forall u,v\in
V,\quad(\varphi,\psi)_W=\int_{\Omega}\nabla\varphi\cdot\nabla
\psi\,dx\quad\forall \varphi,\,\psi\in W.
\]
Moreover, the associated norms
\begin{equation}\label{3.1}
\|v\|_V= \|\nabla v\|_{L^2(\Omega)^2}\quad\forall v\in V,\quad
\|\psi\|_W= \|\nabla\psi\|_{L^2(\Omega)^2}\quad\forall\psi\in W
\end{equation}
are equivalent on $V$ and $W$, respectively, with the usual norm
$\|\cdot\|_{H^1(\Omega)}$. By Sobolev's trace Theorem we deduce
that there exist two positive constants $c_V>0$ and $c_W>0$ such
that
\begin{equation}\label{3.2}
\|v\|_{L^{2}(\Gamma_3)}\leq c_V \|v\|_V \quad\forall
v\in V,\quad \|\psi\|_{L^{2}(\Gamma_3)}\leq c_W\|\psi\|_W
\quad\forall \psi\in W.
\end{equation}
For a real Banach space $(X,\|\cdot\|_X)$ we use the usual
notation for the spaces $L^p(0,T;X)$ and $W^{k,p}(0,T;X$) where
$1\leq p \leq \infty, \ k=1,2,\dots$; we also denote by $C([0,T];
X)$ and $C^1([0,T]; X)$ the spaces of continuous and continuously
differentiable functions on $[0,T]$ with values in $X$, with the
respective norms
\begin{gather*}
\|x\|_{C([0,T]; X)}=\max_{t\in [0,T]} \|x(t)\|_X,
\\
\|x\|_{C^1([0,T]; X)}=\max_{t\in [0,T]} \|x(t)\|_X+\max_{t\in
[0,T]} \|\dot{x}(t)\|_X.
\end{gather*}
Here and subsequently, we still write $w$ for the trace $\gamma w$
of a function $w$ on $\Gamma$, for all $w\in V$.
In the study of the Problem $\mathcal{P}$, we assume that the
viscosity coefficient and the electric permittivity coefficient
satisfy
\begin{gather}\label{3.3}
\theta\in L^\infty(\Omega)\mbox{ and there exists
}\theta^*>0 \mbox{ such that }\ \theta(\mathbf{x})\geq \theta^* \mbox{
a.e. }\mathbf{x}\in \Omega,
\\
\label{3.4}\alpha\in L^\infty(\Omega)\mbox{ and there exists
}\alpha^*>0 \mbox{ such that }\ \alpha(\mathbf{x})\geq \alpha^* \mbox{ a.e.
}\mathbf{x}\in \Omega.
\end{gather}
We also assume that the Lam\'e coefficient and the piezoelectric
coefficient satisfy
\begin{gather}\label{3.5}
\mu\in L^\infty(\Omega)\quad\mbox{and} \quad
\mu(\mathbf{x})>0\ \mbox{ a.e. }\mathbf{x}\in \Omega,
\\
\label{3.6} e\in L^\infty(\Omega).
\end{gather}
The forces, tractions, volume and surface free charge densities
have the regularity
\begin{gather}\label{3.7}
f_0\in W^{1,2}(0,T;L^2(\Omega)), \quad f_2\in
W^{1,2}(0,T; L^2(\Gamma_2)),\\
\label{3.8}q_0\in W^{1,2}(0,T;L^2(\Omega)), \quad q_2\in
W^{1,2}(0,T; L^2(\Gamma_b)).
\end{gather}
The electric conductivity coefficient
satisfy
\begin{gather}\label{3.9}
k\in L^\infty(\Gamma_3)\quad\mbox{and}\quad k(\mathbf{x})\geq
0\ \mbox{ a.e. }\mathbf{x}\in \Gamma_3.
\end{gather}
Finally, we assume that the electric potential of the foundation
and the initial displacement are such that
\begin{gather}\label{3.10}
\varphi_F\in W^{1,2}(0,T;L^2(\Gamma_3)),
\end{gather}
We suppose that the friction bound function $g$ satisfies the
following properties:
\begin{equation} \label{3.11}
\begin{aligned}
&(a)\,\ g :{\Gamma_{3}}\times\mathbb{R}\to\mathbb{R_{+}};\\
&(b)\,\ \exists L_{g}\geq 0\mbox{ such that } |g(x,r_{1})- g(x,r_{2})|
\leq L_{g}|r_{1}-r_{2}|, \\
&\quad \forall\, r_{1},r_{2}\in \mathbb{R}\mbox{ a.e. } x\in\Gamma_{3};\\
&(c)\,\ \forall r\in\mathbb{R}, g(.,r) \mbox{ is Lebesgue measurable on }\Gamma_{3};\\
&(d)\,\ g(.,r) \in L^{2}(\Gamma_{3}).
\end{aligned}
\end{equation}
The initial data are chosen such that
\begin{gather}\label{3.12}
u_{0}\in\,\ V.
\end{gather}
For every $t\in [0,T]$ we need to consider the operator
$\mathcal {S}_t$ defined by
\begin{equation}
\begin{gathered}\label{3.13}
\mathcal {S}_t : L^{\infty}(0,T;V)\to L^{2}(\Gamma),\\
{\mathcal {S}_t}(v) = \int_{0}^{t}\, | v(s)| \, ds\,
\mbox{ a.e. on }\, \Gamma.\\
\end{gathered}
\end{equation}
From (\ref{3.2}) and (\ref{3.13}) it follows that the for
all $v_1, v_2\, \in L^{\infty}(0,T;V)$ the following inequality holds:
\begin{equation} \label{3.14}
\|\mathcal {S}_t(v_1) - \mathcal {S}_t(v_2) \|_{L^{2}(\Gamma)}\leq C
\int_{0}^{t} \| v_1(s)-v_2(s)\|_V \, ds.
\end{equation}
Here and bellow $C$ represents a positive constant whose value
may change from line to line.
We define now the functional
$j : {L^{2}(\Gamma)}\times V\to {\mathbb{R}}_+$ given by
\begin{equation} \label{3.15}
j(\xi,v)=\int_{\Gamma_{3}} g(\xi)|v| \, da\quad \forall
\xi\in {L^{2}(\Gamma)},\,\ \forall v\in V.
\end{equation}
Using conditions (\ref{3.11}), we deduce that the integral
in (\ref{3.15}) is well defined.
We also define the mappings $f : [0,T]\to V$ and $q : [0,T]\to W$
respectively, by
\begin{gather}\label{3.16}
(f(t),v)_V=\int_\Omega f_0(t)v\,dx+\int_{\Gamma_2}
f_2(t)v\,da,\\
\label{3.17}(q(t),\psi)_W=\int_\Omega
q_0(t)\psi\,dx-\int_{\Gamma_b}
q_2(t)\psi\,da+\int_{\Gamma_3}k\,\varphi_F(t)\psi\,da,
\end{gather}
for all $v\in V$, $\psi\in W$ and $t\in[0,T]$. The definition of $f$
and $q$ are based on Riesz's representation theorem; moreover,
it follows from assumptions by \ref{3.7} and \ref{3.8}, that the
integrals above are well-defined and
\begin{equation} \label{3.18}
f\in W^{1,2}(0,T;V),\quad q\in W^{1,2}(0,T;W).
\end{equation}
We define now the bilinear forms :
\begin{gather}\label{3.19}
a_\theta: V\times V\to\mathbb{R}, \quad a_\theta(u,v)
=\int_\Omega\theta\,\nabla u\cdot\nabla v\,dx, \\
\label{3.20}
a_\mu: V\times V\to\mathbb{R}, \quad a_\mu(u,v)=\int_\Omega\mu\,
\nabla u\cdot\nabla v\,dx, \\
\label{3.21}
a_e: V\times W\to\mathbb{R}, \quad a_e(u,\varphi)=\int_\Omega e\,\nabla
u\cdot\nabla\varphi\,dx=a_e^*(\varphi,u),\\
\label{3.22}
a_\alpha: W\times W\to\mathbb{R},\quad a_\alpha(\varphi,\psi)
=\int_\Omega\alpha\,\nabla
\varphi\cdot\nabla\psi\,dx+\int_{\Gamma_3}k\,\varphi\psi\,dx,
\end{gather}
for all $u, v\in V$ and $\varphi, \psi\in W$.
Using the conditions in (\ref{3.15})-- (\ref{3.18}), we deduce that
the integrals (\ref{3.19})-- (\ref{3.22}) are well defined.
From (\ref{3.1})--(\ref{3.2}), we can deduce that the bilinear
forms $a_\theta(\cdot,\cdot)$, $a_\mu(\cdot,\cdot)$, $a_e(\cdot,\cdot)$,
$a_e^*(\cdot,\cdot)$ and $a_\alpha(\cdot,\cdot)$ are continuous;
moreover, the forms $a_\theta(\cdot,\cdot)$, $a_\mu(\cdot,\cdot)$
and $a_\alpha(\cdot,\cdot)$ are symmetric and in addition,
the form $a_\theta(\cdot,\cdot)$ is $V$-elliptic and
$a_\alpha(\cdot,\cdot)$ is $W$-elliptic, since
\begin{gather}\label{3.23}
a_\theta(v,v)\ge \theta^*\|v\|^2_V\quad\forall v\in V,\\
\label{3.24}a_\alpha(\psi,\psi)\ge \alpha^*\|\psi\|^2_W\quad\forall
\psi\in W.
\end{gather}
The variational formulation of Problem $\mathcal{P}$ is based of
the following result.
\begin{lemma}\label{l:1}
If $(u,\varphi)$ is a smooth solution to Problem $\mathcal{P}$,
then $(u(t),\varphi(t))\in V\times W$ and we have:
\begin{gather}\label{3.25}
\begin{aligned}
&a_\theta(\dot{u}(t),v-\dot{u}(t))+a_\mu(u(t),v-\dot{u}(t))+
a_e^*(\varphi(t),v- \dot{u}(t))\\
&+j(\mathcal {S}_t(\dot u),v)- j(\mathcal {S}_t(\dot u),
\dot u(t))\\
&\geq(f(t),v-\dot{u}(t))_V \quad\forall v\in V,\ t\in[0,T],
\end{aligned}\\
\label{3.26}
a_\alpha(\varphi(t),\psi)-a_e(u(t),\psi) =(q(t),\psi)_W
\quad\forall \psi\in W,\ t\in[0,T],\\
\label{3.27}u(0)=u_0.
\end{gather}
\end{lemma}
\begin{proof}
Let $(u,\varphi)$ denote a smooth solution to Problem $\mathcal{P}$,
we have $u(t)\in V$, $\dot u(t)\in V$ and $\varphi(t)\in W$ a.e.
$t\in [0,T]$ and, from (\ref{2.36}), (\ref{2.38}) and (\ref{2.39}),
we obtain
\begin{align*}
&\int_\Omega\theta\,\nabla \dot u(t)\cdot\nabla (v-\dot u(t))\,dx
+ \int_\Omega\mu\,\nabla u(t)\cdot\nabla (v-\dot u(t))\,dx\\
&+ \int_\Omega e\,\nabla \varphi(t)\cdot\nabla (v-\dot u(t))\,dx \\
&= \int_\Omega {f_{0}(t)}\,(v-\dot u(t))\,dx
+ \int_{\Gamma_2}\, {f_{2}(t)}\,(v-\dot u(t))\,da\\
&\quad + \int_{\Gamma_3}\,(\theta\partial_\nu\dot u(t)+\mu\partial_\nu
u(t)+e\partial_\nu\varphi(t))(v-\dot u(t))\,da,\quad\forall
v\in V\,\ t \in (0,T),
\end{align*}
and from (\ref{2.37}) and (\ref{2.41})--(\ref{2.42}) we obtain
\begin{equation}\label{3.28}
\begin{aligned}
&\int_\Omega\alpha\,\nabla\varphi(t)\cdot\nabla\psi\,dx
- \int_\Omega e\,\nabla u(t)\cdot\nabla\psi\,dx \\
&= \int_\Omega q_0(t)\psi\,dx- \int_{\Gamma_b}
q_2(t)\psi\,da+\int_{\Gamma_3}k\,\varphi_F(t)\psi\,da
\quad \forall \psi\in W\,\ t\in (0,T).
\end{aligned}
\end{equation}
Using (\ref{3.16}) and (\ref{3.19})--(\ref{3.21}) we obtain
\begin{equation} \label{3.29}
\begin{aligned}
&a_\theta(\dot{u}(t),v-\dot{u}(t))+a_\mu(u(t),v-\dot{u}(t))
+ a_e^*(\varphi(t),v- \dot{u}(t))\\
& - \int_{\Gamma_3}\,(\theta\partial_\nu\dot u(t)+\mu\partial_\nu
u(t)+e\partial_\nu\varphi(t))(v-\dot u(t))\,da\\
& = (f(t),v-\dot{u}(t))_V ,
\quad\forall v\in V,\ t\in[0,T],
\end{aligned}
\end{equation}
Keeping in mind (\ref{3.17}) and (\ref{3.21})--(\ref{3.22}),
we find the second equality in Lemma \eqref{l:1}; i.e.,
\begin{gather}\label{3.30}
a_\alpha(\varphi(t),\psi)-a_e(u(t),\psi) =(q(t),\psi)_W
\quad\forall \psi\in W,\ t\in[0,T],
\end{gather}
Using the frictional contact condition (\ref{2.40})
and (\ref{3.15}) on $\Gamma_3\times(0,T)$, we deduce that for
all $t\in[0,T]$
\begin{equation} \label{3.31}
j(\mathcal {S}_t(\dot u),\dot u(t))
= - \int_{\Gamma_3}\,(\theta\partial_\nu\dot u(t)+\mu\partial_\nu
u(t)+e\partial_\nu\varphi(t))\dot u(t)\,da,
\end{equation}
it is very easy to see that
\begin{equation} \label{3.32}
j(\mathcal {S}_t(\dot u),v)\geq - \int_{\Gamma_3}\,(\theta\partial_\nu\dot u(t)+\mu\partial_\nu
u(t)+e\partial_\nu\varphi(t)) v\,da,\quad\forall v\in V.
\end{equation}
The first inequality in Lemma \eqref{l:1} follows now from (\ref{3.29}) and (\ref{3.31})--(\ref{3.32}).
Now, from Lemma \eqref{l:1} and the initial condition
(\ref{3.27}) lead to give the following variational Problem:
\subsection*{Problem $\mathcal{PV}$}
Find a displacement field $u:[0,T]\to V$ and an electric potential field
$\varphi:[0,T]\to W$ such that
\begin{gather}\label{3.33}
\begin{aligned}
&a_\theta(\dot{u}(t),v-\dot{u}(t))+a_\mu(u(t),v-\dot{u}(t))+
a_e^*(\varphi(t),v- \dot{u}(t))\\
&+j(\mathcal {S}_t(\dot u),v)- j(\mathcal {S}_t(\dot u),\dot u(t))\\
&\geq(f(t),v-\dot{u}(t))_V, \quad\forall v\in V,\ t\in[0,T],
\end{aligned}\\
\label{3.34} a_\alpha(\varphi(t),\psi)-a_e(u(t),\psi) =(q(t),\psi)_W,
\quad\forall \psi\in W,\ t\in[0,T],\\
\label{3.35}u(0)=u_0.
\end{gather}
\end{proof}
\section{An existence and uniqueness result}\label{s:4}
Our main existence and uniqueness result, which we state and prove in this section, is the
following.
\begin{theorem}\label{t:1}
Assume that \eqref{3.3}--\eqref{3.12} hold. Then the variational
problem $\mathcal{PV}$ possesses a unique solution $(u,\varphi)$
satisfies
\begin{equation}\label{reg}
u\in W^{1,2}(0,T;V),\quad \varphi\in W^{1,2}(0,T;W).
\end{equation}
\end{theorem}
A couple of functions $(u,\varphi)$ which solves Problem
$\mathcal{PV}$ is called a weak solution of the antiplane contact
Problem $\mathcal{P}$. We conclude by Theorem \eqref{t:1} that
the antiplane contact Problem $\mathcal{P}$ has a unique weak solution,
provided that \eqref{3.3}--\eqref{3.12} hold.
We turn to the proof of Theorem \eqref{t:1} which will be carried
out in several steps. To this end, in the rest of this section
we assume that \eqref{3.3}--\eqref{3.12} hold and let $\eta$
and $\xi$ be two elements of $W^{1,2}(0,T;V)$.
We consider the following variational Problem:
\subsection*{Problem $\mathcal{PV}^1_{\eta\xi}$}
Find $v_{\eta\xi}:[0,T]\to V$ such that
\begin{gather}\label{4.1}
\begin{aligned}
&a_\theta(v_{\eta\xi}(t),v-v_{\eta\xi}(t))
+(\eta(t),v-v_{\eta\xi}(t))_V+j(\mathcal {S}_t(\xi),v)
- j(\mathcal {S}_t(\xi),v_{\eta\xi}(t))\\
&\geq(f(t),v-v_{\eta\xi}(t))_V,
\quad\forall v\in V,\,\ a.e.\,\ t\in[0,T],
\end{aligned}
\end{gather}
The unique solvability of the intermediate Problem
$\mathcal{PV}^1_{\eta\xi}$ follows from the following result:
\begin{lemma}\label{l:2}
There exists a unique solution $v_{\eta\xi}$ to
Problem $\mathcal{PV}^1_{\eta\xi}$. Moreover,
\begin{gather}\label{4.3}
v_{\eta\xi}\in W^{1,2}(0,T;V).
\end{gather}
\end{lemma}
\begin{proof}
It follows from classical results for variational inequalities
that there exists a unique solution $v_{\eta\xi}\in V$
that solves \eqref{4.1} a.e. $t\in(0,T)$.
Taking $v=0_V$ in \eqref{4.1}, we deduce that
\[
a_\theta(v_{\eta\xi}(t),-v_{\eta\xi}(t))
+ (\eta(t),-v_{\eta\xi}(t))_V \geq(f(t),-v_{\eta\xi}(t))_V,
\quad a.e.\,\ t\in[0,T].
\]
From \eqref{3.23}, we can deduce that
\begin{equation} \label{4.4}
\theta\|v_{\eta\xi}(t)\|_V \leq \|f(t)\|_V
+ \|\eta(t)\|_V,\quad \text{a.e. } t\in[0,T],
\end{equation}
Taking in mind (\ref{4.4}), (\ref{3.18}) and the regularity
$\eta\in W^{1,2}(0,T;V)$, we obtain $v_{\eta\xi}\in W^{1,2}(0,T;V)$,
which conclude the proof.
\end{proof}
In the next step, we use the displacement field $v_{\eta\xi}$
obtained in Lemma \eqref{l:2} to define the following variational
Problem $\mathcal{PV}^2_{\eta\xi}$ for the electrical potential field:
\subsection*{Problem $\mathcal{PV}^2_{\eta\xi}$}
Find an electrical potential field $\varphi_{\eta\xi}:[0,T]\to V$
such that
\begin{gather}\label{4.5}
a_\alpha(\varphi_{\eta\xi}(t),\psi)+a_e(u_{\eta\xi}(t),\psi)
= (q(t),\psi)_W,
\quad\forall \psi\in W,\,\ t\in[0,T],
\end{gather}
The unique solvability of the electrical Problem
$\mathcal{PV}^2_{\eta\xi}$ follows from the following result:
\begin{lemma}\label{l:3}
There exists a unique solution $\varphi_{\eta\xi}$ to
Problem $\mathcal{PV}^2_{\eta\xi}$ such that
\begin{equation}\label{4.6}
\varphi_{\eta\xi}\in W^{1,2}(0,T;W),
\end{equation}
which satisfies \eqref{4.5}, Moreover, if $\varphi_{{\eta\xi}_1}$
and $\varphi_{{\eta\xi}_2}$ are the solutions of \eqref{4.5}
corresponding to ${\eta\xi}_1$, ${\eta\xi}_2$ $\in C([0,T];V)$ then,
there exists $c>0$, such that
\begin{gather}\label{pot}
\|\varphi_{{\eta\xi}_1}(t)-\varphi_{{\eta\xi}_2}(t)\|_W\leq
c \|u_{{\eta\xi}_1}(t)-u_{{\eta\xi}_2}(t)\|_V.
\end{gather}
\end{lemma}
\begin{proof}
Let $t\in[0,T]$. We use the properties of
the bilinear form $a_\alpha$ and the Lax-Milgram Lemma to see that
there exists a unique element $\varphi_{\eta\xi}(t)\in W$ which solves
$\mathcal{PV}^2_{\eta\xi}$ at any moment $t\in[0,T]$. Consider now $t_1,\,
t_2\in[0,T]$; using (\ref{3.24}) and (\ref{4.5}) we find that
\begin{equation} \label{4.7}
\begin{aligned}
\alpha^* \|\varphi(t_1)-\varphi(t_2)\|_W^2
& \leq \|e\|_{L^\infty(\Omega)}\|u(t_1)-u(t_2)\|_V \|\varphi(t_1)
-\varphi(t_2)\|_W\\
&\quad + \|q(t_1)-q(t_2)\|_W \|\varphi(t_1)-\varphi(t_2)\|_W
\end{aligned}
\end{equation}
which implies
\begin{equation} \label{4.8}
\|\varphi(t_1)-\varphi(t_2)\|_W\leq
c\,(\|u(t_1)-u(t_2)\|_V+ \|q(t_1)-q(t_2)\|_W).
\end{equation}
We note that regularity $u_{\eta\xi} \in C^1( [0,T];V)$ combined
with (\ref{3.18}) and (\ref{4.8}) imply that
$\varphi_{\eta\xi}\in W^{1,2}(0,T;W)$, which concludes the proof.
\end{proof}
We consider now the operator $\Lambda_\eta : C( [0,T]; V)\to C( [0,T]; V)$
defined for all $\eta\in L^{\infty}(0,T;V)$ by the equality
\begin{gather}\label{4.9}
(\Lambda_\eta\xi(t),w)_V=a_\mu(v_{\eta\xi}(t),w)
+a^*_e(\varphi_{\eta\xi}(t),w) \quad\forall w\in V,\,\ t\in [0,T].
\end{gather}
We have the following result.
\begin{lemma}\label{l:4}
For every element $\eta\in L^{\infty}(0,T;V)$ the operator $\Lambda_\eta$ has a unique fixed point $\xi_\eta\in L^{\infty}(0,T;V)$.
\end{lemma}
\begin{proof}
Let $\eta\in L^{\infty}(0,T;V)$ and $\xi_i\in L^{\infty}(0,T;V)$, $i=1,2$.
To simplify the notation, we denote by $v_i$ the unique solution
to Problem $\mathcal{PV}^1_{{\eta\xi}_i}$, for $i=1,2$.
Thus, from (\ref{4.1}) we can write
\begin{equation}\label{4.10}
\begin{aligned}
&a_\theta(v_{i}(t),v-v_{i}(t))+(\eta(t),v-v_{i}(t))_V
+j(\mathcal {S}_t(\xi_i),v)- j(\mathcal {S}_t(\xi_i),v_{i}(t))\\
&\geq(f(t),v-v_{i}(t))_V,
\quad\forall v\in V,\quad\text{a.e. } t\in[0,T].
\end{aligned}
\end{equation}
After some algebra and from (\ref{4.10}), we find
\begin{equation}\label{4.11}
\begin{aligned}
\theta\|v_1(t)-v_2(t)\|^2_V
&\leq j(\mathcal {S}_t(\xi_1),v_{2}(t))
-j(\mathcal {S}_t(\xi_2),v_{2}(t))\\
&\quad +j(\mathcal {S}_t(\xi_2),v_{1}(t))-j(\mathcal {S}_t(\xi_1),v_{1}(t)),\,\ a.e.\,\ t\in[0,T].
\end{aligned}
\end{equation}
Using now (\ref{3.2}), (\ref{3.11}), (\ref{3.14}) and (\ref{3.15}),
it follows that
\begin{equation} \label{4.12}
\begin{aligned}
&j(\mathcal {S}_t(\xi_1),v_{2}(t))-j(\mathcal {S}_t(\xi_2),
v_{2}(t))+j(\mathcal {S}_t(\xi_2),v_{1}(t))
-j(\mathcal {S}_t(\xi_1),v_{1}(t))\\
&\leq C\times \|v_1(t)-v_2(t)\|_V
\int_{0}^{t} \| \xi_1(s)-\xi_2(s)\|_V \, ds\quad\text{a.e. }
t\in[0,T].
\end{aligned}
\end{equation}
Using (\ref{4.11}), (\ref{4.12}) we deduce that
\begin{gather}\label{4.13}
\|v_1(t)-v_2(t)\|_V\leq C \int_{0}^{t} \| \xi_1(s)
-\xi_2(s)\|_V \, ds,\quad \text{a.e. } t\in[0,T].
\end{gather}
Let $\eta_1$, $\eta_2\in C([0,T];V)$ and denote by $u_i$ and
$\varphi_i$ the functions $u_{\eta\xi_i}$ and
$\varphi_{\eta\xi_i}$ obtained in Lemmas \ref{l:2} and \ref{l:3},
for $i=1,2$. Let $t\in [0,T]$. Using the definition \eqref{4.9}
of the operator $\Lambda_\eta$ we obtain
\begin{equation} \label{Lam1}
(\Lambda_\eta\xi_1(t),w)_V=a_\mu(v_{1}(t),w)+a^*_e(\varphi_{1}(t),w)
\quad\forall w\in V,\,\ t\in [0,T],
\end{equation}
and
\begin{equation} \label{Lam2}
(\Lambda_\eta\xi_2(t),w)_V=a_\mu(v_{2}(t),w)+a^*_e(\varphi_{2}(t),w)
\quad\forall w\in V,\,\ t\in [0,T].
\end{equation}
From \eqref{3.21}, \eqref{4.5}, \eqref{Lam1} and \eqref{Lam2} we deduce
\[
\|\Lambda_\eta\xi_1(t)-\Lambda_\eta\xi_2(t)\|_V\leq C\,(\int_{0}^{t} \|v_1(s)-v_2(s)\|_V \, ds +
\|\varphi_1(t)-\varphi_2(t)\|_W),
\]
and, keeping in mind (\ref{pot}) and (\ref{4.13}), we find
\begin{equation}\label{4.14}
\|\Lambda_\eta\xi_1(t)-\Lambda_\eta\xi_2(t)\|_V\leq
C'\int_{0}^{t} \| \xi_1(s)-\xi_2(s)\|_V \, ds,\quad \text{a.e. }
t\in[0,T].
\end{equation}
We define now the set $\|\cdot\|_\kappa$ as follows
\begin{equation}\label{4.15}
\|v\|_\kappa = Inf \left\{ M>0\,\ e^{-\kappa t}\|v\|_V
\leq M\quad \text{a.e. } t\in (0,T) \right\},\quad\forall
v\in L^{\infty}(0,T;V),
\end{equation}
such that $\kappa > 0$ to be determined later. The norm
$\|\cdot\|_\kappa$ is equivalent to the standard norm
$\|\cdot\|_{L^{\infty}(0,T;V)}$. Using now (\ref{4.15})
and the definition of the norm $\|\cdot\|_\kappa$, we can obtain :
\begin{equation}\label{4.16}
e^{-\kappa t} \|\Lambda_\eta\xi_1(t)-\Lambda_\eta\xi_2(t)\|_V\leq C'\,
e^{-\kappa t}\int_{0}^{t}\, e^{\kappa s}e^{-\kappa s}\| \xi_1(s)
-\xi_2(s)\|_V \, ds,
\end{equation}
then
\begin{equation}\label{4.17}
e^{-\kappa t} \|\Lambda_\eta\xi_1(t)-\Lambda_\eta\xi_2(t)\|_V\leq C'\, e^{-\kappa t} \|\xi_1-\xi_2\|_\kappa\int_{0}^{t}\,e^{\kappa s}\, ds,
\end{equation}
we deduce
\begin{equation}\label{4.18}
e^{-\kappa t} \|\Lambda_\eta\xi_1(t)-\Lambda_\eta\xi_2(t)\|_V\leq \frac{C'}{\kappa}\, e^{-\kappa t} \|\xi_1-\xi_2\|_\kappa,\quad a.e.\,\ t\in[0,T].
\end{equation}
Consequently, we deduce that
\begin{equation}\label{4.19}
\|\Lambda_\eta\xi_1(t)-\Lambda_\eta\xi_2(t)\|_\kappa\leq\frac{C'}{\kappa} \| \xi_1-\xi_2\|_\kappa.
\end{equation}
Taking $\kappa$ such that $\kappa > C'$, we conclude that the
operator $\Lambda_\eta$ is a contraction on the space
$(L^{\infty}(0,T;V),\|\cdot\|_\kappa)$. From Banach fixed point
Theorem, we deduce that the operator $\Lambda_\eta$ has a
unique fixed-point $\xi_\eta\in L^{\infty}(0,T;V)$.
\end{proof}
In what follows, we continue to write
\begin{equation}\label{4.20}
v_\eta = {v_{\eta\xi}}_\eta, \quad \forall \eta\in L^{\infty}(0,T;V),
\end{equation}
where $\xi_\eta$ is the unique fixed point of the operator $\Lambda_\eta$, then from (\ref{4.9}) and (\ref{4.20}) we obtain
\begin{equation}\label{4.21}
v_\eta = \xi_\eta.
\end{equation}
Let $u_\eta : [0,T]\to V$ be the function defined by
\begin{equation}\label{int}
u_\eta = \int_{0}^{t}\,v_\eta(s)\, ds +u_0\quad \forall t\in [0,T].
\end{equation}
We also define the operator
$\Lambda : L^{\infty}(0,T;V)\to L^{\infty}(0,T;V)$ by
\begin{equation}\label{opet}
(\Lambda\eta(t),w)_V=a_\mu(u_{\eta}(t),w)+a^*_e(\varphi_{\eta}(t),w)
\quad\forall w\in V,\,\ t\in [0,T].
\end{equation}
We have the following result.
\begin{lemma}\label{l:5}
The operator $\Lambda$ has a unique fixed point
$\eta^*\in L^{\infty}(0,T;V)$.
\end{lemma}
\begin{proof}
Using (\ref{4.21}) and the fact that $v_\eta$ is the unique solution
of Problem $\mathcal{PV}^1_{\eta\xi}$, we can obtain
\begin{equation} \label{4.23}
\begin{aligned}
&a_\theta(v_{\eta}(t),v-v_{\eta}(t))+j(\mathcal {S}_t(v_\eta),v)
- j(\mathcal {S}_t(v_\eta),v_{\eta}(t))\\
&\geq (f(t),v-v_{\eta}(t))_V - (\eta(t),v-v_{\eta}(t))_V,
\quad\forall v\in V,\quad\text{a.e. } t\in[0,T].
\end{aligned}
\end{equation}
Let $\eta_1$, $\eta_2\in L^{\infty}(0,T;V)$ and
denote by $u_i$ and $\varphi_i$ the functions $u_{\eta_i}$ and
$\varphi_{\eta_i}$ obtained in Lemmas \ref{l:2} and \ref{l:3}, for
$i=1,2$. Let $t\in [0,T]$. Using (\ref{4.23}) we obtain
\begin{equation}\label{4.24}
\begin{aligned}
&\|v_1(s)-v_2(s)\|_V\\
&\leq C\, (\|\eta_1(s)-\eta_2(s)\|_V + \int_{0}^{t}
\| v_1(r)-v_2(r)\|_V \, dr), \quad \text{a.e. } s\in[0,T].
\end{aligned}
\end{equation}
We integrate the \eqref{4.24} on the $[0,T]$ with a fixed $t$
and using a Gronwall-type argument, we obtain
\begin{equation} \label{4.25}
\int_{0}^{t} \| v_1(s)-v_2(s)\|_V \, ds \leq C\int_{0}^{t}
\|\eta_1(s)-\eta_2(s)\|_V ds,\,\forall t\in[0,T].
\end{equation}
Using the definition \eqref{4.23} for $\eta_1$ and $\eta_2$ we obtain
\begin{gather}\label{4.26}
(\Lambda\eta_1(t),w)_V=a_\mu(u_{1}(t),w)+a^*_e(\varphi_{1}(t),w)\quad
\forall w\in V,\; t\in [0,T], \\
\label{4.27}
(\Lambda\eta_2(t),w)_V=a_\mu(u_{2}(t),w)+a^*_e(\varphi_{2}(t),w)
\quad\forall w\in V,\; t\in [0,T].
\end{gather}
From \eqref{4.26}--\eqref{4.27} we can write
\begin{equation}\label{4.28}
\|\Lambda\eta_1(t)-\Lambda\eta_2(t)\|_V \leq C (\|u_1(t)
- u_2(t)\|_V + \|\varphi_1(t) - \varphi_2(t)\|_W).
\end{equation}
On the other hand, \eqref{4.5} and arguments similar as those
used in the proof of \eqref{4.9} yield
\begin{equation}\label{4.29}
\|\varphi_1(t) - \varphi_2(t)\|_W \leq c \|u_1(t) - u_2(t)\|_V.
\end{equation}
Using \eqref{int}, \eqref{opet}, \eqref{4.28} and \eqref{4.29} we obtain
\begin{equation}\label{4.30}
\|\Lambda\eta_1(t)-\Lambda\eta_2(t)\|_V \leq C \int_{0}^{t} \|\eta_1(s)
- \eta_2(s)\|_V\, ds,\quad\forall t\in[0,T].
\end{equation}
Keeping in mind the defintion of $\|\cdot\|_\kappa$, Lemma \ref{l:5}
follows from the previous inequality, after using a fixed-point
argument similar to that presented Lemma \ref{l:4}.
\end{proof}
Now we have all the ingredients to prove Theorem \ref{t:1}.
\subsection*{Existence}
Let $\eta^*\in L^{\infty}(0,T;V)$ be the unique fixed point of the
operator $\Lambda$ and let $u$ and $\varphi$ be the solutions of
Problems $\mathcal{PV}^1_{\eta\xi}$ and $\mathcal{PV}^2_{\eta\xi}$
respectively with $\eta=\eta^*$, i.e. $u=u_{\eta^*}$ and
$\varphi=\varphi_{\eta^*}$. Clearly, equalities
\eqref{3.33}--\eqref{3.35} hold from $\mathcal{PV}^1_{\eta\xi}$
and $\mathcal{PV}^2_{\eta\xi}$. Let $u_{\eta^*}\in W^{1,2}(0,T;V)$ be
the function defined by the relation \eqref{int} for $\eta=\eta^*$.
We have $\dot u_{\eta^*} = v_{\eta^*}$ and from \eqref{4.23},
it follows that
\begin{gather}\label{4.31}
\begin{aligned}
&a_\theta(\dot u_{\eta^*}(t),v-\dot u_{\eta^*}(t))+j
(\mathcal {S}_t(\dot u_{\eta^*}),v)- j(\mathcal {S}_t(\dot u_{\eta^*}),
\dot u_{\eta^*}(t))\\
&\geq (f(t),v-\dot u_{\eta^*}(t))_V
- (\eta^*(t),v-\dot u_{\eta^*}(t))_V,
\quad\forall v\in V,\quad\text{a.e. } t\in[0,T].
\end{aligned}
\end{gather}
The inequality \eqref{3.33} follows now from \eqref{4.31}
and \eqref{opet}, using the fact that $\eta^*$ is the fixed
point of the operator $\Lambda$. From the definition \eqref{int}
implies $u_{\eta^*}(0) = u_0$ so that \eqref{3.35} is fulfilled.
We conclude now that $u_{\eta^*}$ is a solution to Problem $\mathcal{PV}$.
The regularity of the solution expressed in \eqref{reg} follows
from Lemmas \ref{l:2} and \ref{l:3}. We conclude that $(u,\varphi)$
is a solution of Problem $\mathcal{PV}$ and it satisfies \eqref{reg}.
\subsection*{Uniqueness}
The uniqueness of the solution follows from the uniqueness of the
fixed point of $\Lambda$ combined with the unique solvability of
previous Problems, guaranteed by Lemmas \ref{l:2}--\ref{l:4}.
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\end{document}